Fractional Insights in Tumor Modeling: An Interactive Study Between Tumor Carcinogenesis and Macrophage Activation

Abstract Biological studies have shown that macrophages play a dual role in cancer dynamics. Through the secretion of different factors, they can modulate the tumor microenvironment and directly interact with tumor cells. Active macrophages, more precisely tumor‐associated macrophages (TAM), can secrete substances that increase the proliferation and survival of tumor cells, thus promoting tumor growth. This study explores the dynamic interactions between tumor carcinogenesis and macrophage activation through a newly developed mathematical model using the Caputo sense fractional operator. The main focus is to analyze the influence of macrophage polarization on tumor progression and cancer development. To precisely capture the dynamic behavior of macrophage‐tumor interactions, the model is verified for qualitative analysis. The fixed points of the model are obtained and their stability analysis is carried out. It is started by explaining the essential components of the model and then assuring its mathematical reliability. The Adams–Bashforth–Moulton method (ABM) is also discussed to show its applicability and efficiency for the numerical results of the developed model. The findings shed light on cancer progress, potential analyses, and the influence of public information. The outcome of the study offers a more detailed understanding of possible treatment targets by highlighting the delicate balance between macrophage pro‐ and anti‐tumor effects. The outcomes might improve public health policy, healthcare budget apportionment, and contest tumors and related infections. Further, it improves the understanding of tumor‐immune system interactions and lays the solid basis for cancer treatment research. Finally, numerical simulations make our outcomes more significant.